TY - JOUR
T1 - Pathologies of the large- N limit for RPN-1 , CPN-1 , QPN-1 and mixed isovector/isotensor σ -models
AU - Sokal, Alan D.
AU - Starinets, Andrei O.
N1 - Funding Information:
We wish to thank Paolo Butera, Sergio Caracciolo, Hervé Kunz, Andrea Pelissetto and Paolo Rossi for many helpful discussions, and Larry Glasser for correspondence. In particular, it was Paolo Butera who pointed out to one of us (A.D.S.), nearly a decade ago, that the N=∞ phase transition in RP N−1 models occurs also in dimension d=1 — a comment that directly inspired the present work. This research was supported in part by U.S. National Science Foundation grants PHY-9520978 and PHY-9900769. It was also aided by the gracious hospitality of the Scuola Normale Superiore.
PY - 2001/5/14
Y1 - 2001/5/14
N2 - We compute the phase diagram in the N→∞ limit for lattice RPN-1, CPN-1 and QPN-1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.
AB - We compute the phase diagram in the N→∞ limit for lattice RPN-1, CPN-1 and QPN-1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.
KW - Mixed isovector/isotensor model
KW - Nematic liquid crystal N→∞ limit 1/N expansion
KW - Nonlinear σ -model
KW - RP model CP model QP model N -vector model
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U2 - 10.1016/S0550-3213(01)00065-7
DO - 10.1016/S0550-3213(01)00065-7
M3 - Article
AN - SCOPUS:0035858799
SN - 0550-3213
VL - 601
SP - 425
EP - 502
JO - Nuclear Physics B
JF - Nuclear Physics B
IS - 3
ER -